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Green Noise Digital Halftoning with Multiscale Error Diffusion

Yik-Hing Fung and Yuk-Hee Chan†

Center of Multimedia Signal Processing Department of Electronic and Information Engineering The Hong Kong Polytechnic University, Hong Kong

ABSTRACT

Multiscale error diffusion (MED) is superior to conventional error diffusion algorithms as it can eliminate

directional hysteresis completely and possesses a good blue noise characteristic. However, due to its filter design,

it is not suitable for systems with poor isolated dot generation and instable dot gain. In this paper, we propose a

multiscale error diffusion algorithm to produce halftones of desirable green noise characteristics. This algorithm

allows one to adjust the desirable cluster size freely through a single parameter and supports a linear relationship

between the cluster size and the input gray level. With a close-to-isotropic diffusion filter, the algorithm can

effectively remove pattern artifacts, eliminate directional artifacts and preserve original image details. Analysis

and simulation results show that it provides better performance in terms of various aspects including dot

distribution, anisotropy and output image quality as compared with other conventional green noise error diffusion

algorithms.

EDICS: ELI-PRT Index Terms – Multiscale error diffusion (MED), error diffusion, halftoning, multiscale processing, printing,

green noise.

† Corresponding author (Tel: (852)27666264; Fax: (852)23628439; Email: enyhchan@polyu.edu.hk)

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I. INTRODUCTION

Digital halftoning is a technique used to turn a gray-level image into a bi-level image and has been widely

used in printing applications [1]. Basically, halftoning can be accomplished in two ways. One is amplitude

modulation(AM) [2] in which a halftone is produced by varying the size of printed dots arranged along a regular

grid. The other one is frequency modulation(FM) [3-5] in which a halftone is produced by varying the relative dot

density of fixed-size printed dots.

As compared with AM halftoning, FM halftoning produces halftones of higher spatial resolution and better

image quality[4]. Unlike outputs of AM halftoning, halftones produced by FM halftoning are free of moiré

artifacts as neighboring pixels have been taken into account during error diffusion.

However, halftones produced by FM halftoning are sensitive to the dot gain of a printer, which is the increase

in size of the printed dot relative to the intended dot size. In practice, the size and the shape of a printed dot are

not as perfect as they are expected and the printed halftone generally appears darker than expected. When the

variation in dot size and shape from printed dot to printed dot is small, printing can rely on dot gain compensation

technique to minimize this distortion [6]. However, dot gain compensation does not work effectively when the

reliability to produce isolated dots is not stable. In such a case, other than using AM halftoning, using a halftoning

method which produces clustered dots helps to reduce the dot gain distortion as a cluster has a lower perimeter to

area ratio as compared with an isolated dot.

AM-FM halftoning is a hybrid version of AM and FM halftoning which aims at producing clustered dots. For

example, Levien [7] proposed an output-dependent feedback error diffusion algorithm to form dot clusters of

adjustable size to increase the printing stability. As a result, the output halftone image can be tuned to have larger

cluster dots in a high dot gain situation to cater for different printing systems.

Through a systematic study on the halftones produced with various algorithms, Lau found that a halftone

bearing green noise characteristics which contains only mid-frequency spectral components is less susceptible to

image degradation from non-ideal printing device [8]. Based on Lau’s idea, Damera-Venkata et al. proposed an

adaptive threshold modulation framework to improve the halftone quality by optimizing error diffusion

parameters in the least squares sense and derived an adaptive algorithm to provide halftones of green noise

characteristics [9]. A block error diffusion algorithm which allows a user to determine the dot size and shape

directly was later proposed by the same research group [10].

Multiscale error diffusion (MED) was first proposed in [11] and then extensively improved in [12-15]. It has

already been proven to be superior to conventional error diffusion algorithms in terms of various measuring

criteria[14-15]. The comprehensive empirical analysis provided in [14] shows that the outputs of MED algorithms

are free of directional hysteresis and possess good blue noise characteristics. Its success in providing a blue-noise

halftone motivates us to explore whether the MED framework can also be used in green noise halftoning which

aims at providing halftones of green noise characteristics.

In this paper, a MED algorithm is proposed to produce halftones of desirable green noise characteristics. This

algorithm allows one to adjust the desirable cluster size freely through a single parameter and supports a linear

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relationship between the cluster size and the input gray level. With a proposed close-to-isotropic diffusion filter,

the algorithm effectively removes pattern artifacts, eliminates directional artifacts and preserves original image

details. Simulation results show that it provides better performance in terms of various aspects including dot

distribution, anisotropy and visual image quality as compared with other conventional green noise error diffusion

algorithms.

The organization of this paper is as follows. In section II, we briefly review a MED algorithm and introduce

the motivation of our proposal. In section III, diffusion filters are redesigned for MED algorithms to achieve green

noise halftoning. In section IV, how the parameters of the suggested diffusion filter affects the average cluster size

of the halftoning output is addressed. Section V provides a detailed empirical analysis on the performance of the

proposed algorithm in terms of various spatial and spectral statistics. In section VI, simulation results on real

images are provided to evaluate the performance of various green noise error diffusion algorithms. The

complexity issue is addressed in Section VII. Finally, a conclusion is given in section VIII.

II. REVIEW OF FMED

In this section, we first briefly review the feature-preserving MED algorithm (FMED) proposed in [13]. This

MED algorithm serves as a typical example of conventional MED algorithms and forms the basis of the proposed

algorithm. Then, we explain why conventional MED algorithms [12-14] do not provide halftones of green noise

characteristics and suggest a modification to make it.

Without loss of generality, consider we want to halftone an input gray-level image X of size ll 22 × , where l

is a positive integer, to obtain an output binary image B. The values of X are within 0 and 1. Here we assume that

the maximum and the minimum intensity values are, respectively, 1 and 0. Note that there is no limitation of the

input image size when using MED algorithms. The mentioned size is for easier illustration only.

FMED is a two-step iterative algorithm. At the beginning, an error image E is initialized to be the gray-level

input image X. Pixels of B are then picked iteratively to determine their intensity values until the termination

criterion is satisfied. For reference purpose, the intensity values of pixels (m,n) of E and B are, respectively,

denoted as em,n and bm,n.

In the first step of each iteration cycle, a pixel in B is selected via the “extreme error intensity guidance” based

on the most updated E. Let the coordinates of the selected pixel be (i,j). A dot is then introduced to pixel (i,j) of B

in the second step by assigning a corresponding value (0 or 1) to bi,j, and the error image E is updated by diffusing

the error (= bi,j - ei,j) to pixel (i,j)’s neighbors in E. The value of ei,j is reset to 0 after the diffusion. Figure 1 briefly

summarizes how FMED operates in an iteration cycle. These two steps are repeated until the sum of all pixels of

E is bounded in absolute value by 0.5. One may refer to [13] for the details of FMED.

The output of FMED bears good blue-noise characteristics [14], which is good to a stable printing situation.

However, when the dot gain is high and dots cannot be consistently reproduced, clustered dots are preferred in the

halftoning output to compensate for the dot-gain distortion.

As a matter of fact, conventional MED algorithms [12-14] are prone to produce isolated dots as they all use

small diffusion filters (3×3 in most cases). After (i,j) is located and bi,j is determined by quantizing ei,j, the

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quantization error (bi,j - ei,j) is diffused to (i,j)’s nearest neighboring pixels in E. This encourages the formation of

isolated dots.

As an example, when one puts a black dot to pixel (i,j) by assigning bi,j to 0, an error diffusion with a 3×3

diffusion filter increases the intensity values of pixels (i,j±1), (i±1,j±1) and (i±1,j) unless ei,j is already 0 before the

diffusion. This intensity increase increases the likeliness of assigning white dots to bi,j±1, bi±1,j and bi±1,j±1 in the

future. Accordingly, it is more likely that the black dot at (i,j) will be surrounded by white dots in the final

halftoning output.

If it is desirable to form a cluster of dots for pixel (i,j), the error should not be diffused to the closest

neighboring pixels with a 3×3 filter. Instead, it should be diffused to the pixels that are farther away from pixel

(i,j). By doing so, the intensity values of the closest neighboring pixels are not affected by the diffusion and hence

it does not directly affect the chance of assigning a particular type of dots to them after the diffusion. In contrast,

the intensity values of the outer neighboring pixels to which the error is diffused are increased, which increases

these pixels’ chance of having white dots in the future. In view of energy conservation, the number of white dots

appeared in the output should equal to the rounding result of the total energy of the input. As the number of white

dots to be assigned to the output is fixed and now the outer neighboring pixels are more likely to be white, the

inner neighboring pixels are actually more likely to be black indeed. Consequently, it encourages the formation of

a dot cluster centered at pixel (i,j).

Based on the aforementioned observations, the diffusion filter used in FMED is redesigned in this paper such

that halftones of green noise characteristics can be produced with FMED. The details are discussed in Section III.

III. DIFFUSION FILTERS FOR GREEN-NOISE HALFTONING

As mentioned in Section II, diffusing the quantization error of pixel (i,j) to pixel (i,j)’s farther neighbors

instead of its connected neighbors encourages the formation of a dot cluster at (i,j). Based on this idea, a straight

forward approach to make FMED produce dot clusters is to replace the original default 3×3 diffusion filter

1H =[0.5, 1, 0.5; 1, 0, 1; 0.5, 1, 0.5]/6 used in FMED with a (2k+1)× (2k+1) square filter kH defined as

( )

⎩⎨⎧

<<==+⋅

=knandkmif

knorkmifnmSnmhk ||||0

||||)/(1)/1(),(

22

for k=2,3… (1)

where ),( nmhk is the (m,n)th filter coefficient of filter kH and ( )∑=

+=knorm

nmS||||

22 )/(1 is a normalization

factor which makes the sum of all filter coefficients be 1. As an example, we have

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

8/15/14/15/18/15/10005/14/10004/15/10005/18/15/14/15/18/1

3110

2H (2)

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The determination of the filter coefficients is based on the idea that, in an isotropic diffusion process, the

intensity at a particular point away from the source is inversely proportional to the square of the distance from the

source.

Filter kH is k–dependent. Its size is of (2k+1)×(2k+1) pixels. The larger the k value, the larger the filter size is

and the larger the clusters can be produced in the halftone output for a fixed gray-level input. Our simulation

results verified this truth. Specifically, when FMED works with 2H , the average cluster size is 5 pixels for a

constant input of gray-level value 0.5.

Though replacing filter 1H with a filter kH with k>1 can help FMED to produce dot clusters, a better

diffusion filter can yet be designed to further improve the halftoning performance.

As discussed in [13], one of the strengths of MED algorithms is that they can use a non-causal filter to diffuse

the quantization error in a radially symmetrical way to eliminate directional hysteresis. To enhance this strength,

one should select an isotropic diffusion filter to diffuse the error to all directions equally. From that point of view,

square filters kH are hence not ideal and a circular ring filter would be a better alternative to work as a diffusion

filter.

When kH is used, one can change the average cluster size for a particular input gray level by adjusting the

value of k. However, the filter size (2k+1)×(2k+1) and hence the average cluster size is not continuously

adjustable. A filter which allows one to continuously adjust the cluster size easily would be more preferable from

application’s point of view.

A. Ideal circular ring filter in continuous spatial domain

Consider the ideal case that the image spatial domain is continuous. A circular ring filter ),( 21 RRF for diffusing

the error at position (0,0) can be defined as

( ) ( )π)(/),(),(),( 21

2212

RRyxfyxfyxf RR −−= x, y are real numbers (3)

where R1 and R2 are, respectively, the inner and the outer radii of the ring, and

⎪⎩

⎪⎨⎧ ≤+=

elseRyxifyxf k

Rk 01),(

22

for 21, RRRk = (4)

In polar coordinate system, eqn.(3) can be rewritten as

( ) ( )π)(/)(')(')(' 21

2212

RRrfrfrf RR −−= (5)

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where 22 yxr += and ⎩⎨⎧ ≤

=else

Rrifrf k

Rk 0 1

)(' for 21, RRRk = . The Fourier transform of )(' rf is then given

as

πρ

ρπρπρ)(

)2()2()(' 21

22

111212

RRRJRRJRF

−−

= (6)

where ρ is the radial coordinate in the frequency domain and )(1 •J is the Bessel function of the first kind of order

1.

The inner radius R1 of the circular ring filter helps to determine the average cluster size of the clusters formed

when the filter works with FMED to produce a halftone. Roughly speaking, as we shall show in Section IV, when

the ratio of the outer radius R2 to the inner radius R1 closes to 1, the average cluster size for a constant gray level

input can be approximated to be π21gR , where g is the gray level value. Conceptually, this can be interpreted as

that, on average, FMED encircles an area of π21R in a constant gray level input (=g) and then puts a white cluster

(=1) of size Mg on a black background in that area to emulate the intensity distribution of the area in its output as

shown in Figure 2. By energy conservation it results in π21gRM g = . One can hence make use of parameter R1 to

adjust the desirable cluster size in the halftoning output.

When R2/R1≈1, the ring width of the circular ring filter is small and filter function (3) approximates the

wavefront of the diffusion at radius R1 from a point error source. Theoretically, the narrower the ring width, the

better the approximation to the wavefront is. However, there are some other factors to be considered when

selecting R2 and we will discuss it later.

B. Approximated ring filter in discrete spatial domain

In practice, the spatial domain of a digital image is not continuous and hence the circular ring diffusion filter

),( 21 RRF has to be adjusted to fit the pixel grid. To achieve this, one can approximate filter ),( 21 RRF with oRRF ),( 21

whose filter coefficients are defined as

∫ ∫+

−=

+

−==

5.0

5.0

5.0

5.0),(),(

m

mx

n

ny

o dxdyyxfnmf m, n are integers (7)

In this arrangement, the filter coefficient for a pixel which is (m,n) pixels away from the point source is the area

covered by the circular ring 122

2 RyxR >+≥ in the grid unit associated with that pixel.

By substituting eqn.(3) into eqn. (7), we have

π)(),,(),,(),( 2

122

12

RRRnmARnmAnmf o

−−

= (8)

7

where

∫ ∫+

−=

+

−==

5.0

5.0

5.0

5.0),(),,(

m

mx

n

ny Rk dxdyyxfRnmAk

for 21, RRRk = (9)

is the area covered by circle kRyx ≤+ 22 in pixel (m,n). In practice, )0,0(of must be zero as all error must be diffused away from the source pixel. This can be

achieved by making 2/11 ≥R . Then we have ),0,0( 1RA = ),0,0( 2RA =1 and hence )0,0(of =0.

The details of the computation of ),,( kRnmA can be found in the appendix of this paper. With appropriate ),,( kRnmA values, the filter coefficients of the approximated ring filter o

RRF ),( 21 can be determined by using eqn.

(8). As an example, the approximated version of filter ),( 21 RRF for 8.11 =R and 28.12 =R ≈2.546 is given as

=oF)28.1,8.1(

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000286.00000680.1421.7107.7421.7680.100421.7085.10085.1421.70286.0107.7000107.7286.00421.7085.10085.1421.700680.1421.7107.7421.7680.10000286.0000

1001 (10)

As distortion is introduced in the approximation, the filter support of oRRF ),( 21

is no longer a circular ring and

the error cannot be diffused to all directions equally.

Figure 3 shows the frequency responses of filters )28.1,8.1(F , oF )28.1,8.1( and 2H for comparison. They are all of

comparable filter size. To a certain extent, both 2H and oF )28.1,8.1( are approximated versions of )28.1,8.1(F . By

inspecting their frequency responses, oF )28.1,8.1( is obviously a better approximation to )28.1,8.1(F as compared with

2H . In theory, when a filter is isotropic, in its frequency response the magnitude variance of the frequency

components covered in an annular ring of any radius should be zero as long as the ring width is sufficiently small.

Figure 4a plots the corresponding measures in rings of different radii for different filters. It also shows that oF )28.1,8.1( is better than 2H in a way that it can diffuse the error to all directions more uniformly.

Figure 4a also shows the performance of oF )1.18.1,8.1( × for shedding some idea on what we should consider when

selecting R2. In practice, the narrower the ring width of the original ),( 21 RRF , the more distortion is introduced by

the approximation process governed by eqn.(7) due to the spatial grid constraint and, after the approximation, the

filter support of oRRF ),( 21

deviates more from a perfect circular ring. This can be verified by the fact that the curve

of filter oF )1.18.1,8.1( × deviates more than that of oF )28.1,8.1( from zero in Figure 4a. Figure 4b shows the case when a

larger R1 is used (=2.8), and that oF )28.2,8.2( is closer than oF )1.18.2,8.2( × to be isotropic is even more obvious. From

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that point of view, a larger value of ratio 12 / RR would be better for oRRF ),( 21

to diffuse the error to all directions

equally. However, a larger filter support results in higher computational effort and visually more blurred halftones.

In our proposal, 2/ 12 =RR is suggested. With such an arrangement, the area of the filter support

( ππ 21

22 RR − ) equals to that of the inner region ( π21R ). We will show in section IV why it is suggested.

C. Working with FMED

Theoretically, filter oRRF ),( 21

can work with any MED algorithms [12-14] to realize green noise halftoning. Here,

we use FMED as an example to show how one can modify a MED algorithm to achieve green noise halftoning.

For reference, the modified FMED is referred to as FMEDg hereafter.

Like all MED algorithms, FMEDg is a two-step iterative algorithm. In each iteration cycle, it selects a pixel to

assign a dot in the 1st step and then diffuses the error to update the error plane E in the 2nd step. E is initialized to

be X at the beginning. The 1st step of FMEDg is exactly the same as the 1st step of FMED except that parameters

offx and offy , the random shifts added to the starting searching window, are bounded in set {0, ±1, ±2} instead of

{0, ±1}.

Assume that pixel (i,j) is selected and a white(black) dot is assigned to it by making bi,j =1(0) in the 1st step. In

the 2nd step, the error between bi,j and ei,j is diffused with a pre-selected default filter oRRF ),( 21

or, if necessary, its

rectified version. The R1 value of the default filter is used to select the desirable cluster size for a particular input

gray level. In particular, the error image E is updated as

⎩⎨⎧

Ω∈−−−⋅⋅−−−=

=′),(,,,,

,21

),(/)(),(),(),(0

RRjijinmo

nmnm jnimifsebdjnimfe

jinmife (11)

where ),( 21 RRΩ is the filter support of oRRF ),( 21

,

⎩⎨⎧

=else1

valueaassignedbeenhasif0 ,,

nmnm

bd and

∑Ω∈−−

⋅−−=)2,1(),(

,),(RRjnim

nmo djnimfs (12)

In the case when s=0, we gradually increase the value of R2 by 0.5 to make s≠0 and keep the algorithm

working.

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IV. ANALYSIS OF CLUSTER SIZE

The motivation for green noise halftoning is to produce patterns with adjustable coarseness that can be tuned

to the reliability of a given printer to produce dots consistently. In ideal case, the average size of the resultant

clusters for a particular gray level input should be continuously adjustable and the average cluster size should be

linearly proportional to the input gray level. This section provides an empirical analysis on how parameters 1R

and 2R of filter oRRF ),( 21

affect the cluster size in a halftone image when oRRF ),( 21

works with FMED.

A simulation was first performed to study how FMEDg changes the cluster size according to different gray

levels. In our study, oRRF )2,( 11

was used as the default diffusion filter. For each particular gray level, a constant

gray-level image of size 128×128 was generated and halftoned with FMEDg to produce 50 halftones. The average

cluster size gM̂ in the resultant halftones was then measured.

Figures 5a and 5b, respectively, show the average and the standard derivation of the cluster sizes measured in

the simulation for different combinations of 1R and g when 12 2RR = . For reference, Figure 5c shows the

surface of a model function formulated as

gRM g π21= (13).

The small scale of difference )ˆ( gg MM − shown in Figure 5d verifies that eqn. (13) is an appropriate model to

describe the relationship among the involved items when 12 2RR = . From the model, one can see that the cluster size is proportional to (i) the input gray level g consistently for

different 1R and (ii) the square value of 1R . According to the properties of a given printer, one can adjust 1R to

select the desirable cluster size for a particular gray level input and allow FMEDg to generate clusters the average

size of which is approximately linearly proportional to the input intensity value for other input gray levels.

The simulation results presented earlier shows the case when 12 2RR = . Figure 6 shows the case of some

other combinations of 2R and 1R . One can see that, when the ratio of 2R to 1R becomes large, the approximately

linear relationship between the measured gM̂ and the input gray level g is no longer valid for a large 1R . The

larger the ratio, the more the practical situation deviates from the model given by (13). By considering that a

smaller ratio makes the filter support of oRRF ),( 21

more deviated from a ring as discussed in Section IIIB,

12 2RR = is hence suggested in this paper.

V. EMPIRICAL PERFORMANCE ANALYSIS

The performance of a halftoning algorithm can be quantitatively evaluated by measuring various spatial and

spectral statistics of its halftoning outputs. An empirical analysis was carried out to study the performance of

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FMEDg and the results are reported in this section. For comparison, the performance of some other green noise

error diffusion algorithms was also evaluated.

In our analysis, various green noise error diffusion algorithms were applied to a set of constant gray-level

images of size 256×256 and the dot distributions in their outputs were studied in terms of different statistics.

Levien [7], Damera-Venkata [9], Damera-Venkata [10] and FMEDg were included in the comparison. In the

realization of [7] and [9], serpentine scanning is used and the hysteresis constant H is set to 1.0. The hysteresis

filter and the error filter used in [7] are, respectively, fixed to be [0, 0.6, 0; 0.4, 0, 0] and [0, 0.5, 0; 0.5, 0, 0].

These two filters are adaptive in [9] and they are, respectively, initialized to be [0, 0.6, 0; 0.4, 0, 0] and [0, 0.5, 0;

0.5, 0, 0] as well. In the realization of FMEDg, filter oF )28.1,8.1( is used as the default filter. The settings of [7], [9]

and FMEDg are selected to make them produce clusters of comparable size (≈5.6 pixels) at the output when the

input gray level is 0.5 for fair comparisons. The dot shape used in simulating [10] is a 2×2 cluster (=[1, 1; 1, 1]). It

is selected because in [10] the allowable default cluster size must be an integer. A cluster of 2×2 dots is of a shape

close to isotropic while its size is close to 5.6 pixels.

Figure 7 shows some portions of the halftoning results for g = 33/255, 60/255, 82/255 and 116/255. The

selected gray levels represent different ranges of input gray levels.

A. Spatial statistics

In [8], Lau developed a directional distribution function )(21, αrrD to measure the directional distribution of

dots in a dot pattern. It is defined as the expected number of dots per unit area in an angular segment of the ring

bounded by an inner radius 1r and an outer radius 2r in the spatial domain. The annular ring is centered at the

center pixel of a cluster and the segment is indexed by α which specifies the segment’s directional position with

respect to the center. In ideal case, we have 1)(21, =αrrD for all α’s, which indicates an isotropic distribution in

the pattern.

To report the directional dot distribution in an intra-cluster region and that in an inter-cluster region separately,

)(,0 αΔD for α={0,π/4,…7π/4} and )(, αλ Δ+Δ gD for α={0,π/8,…15π/8} are, respectively, provided in our analysis.

Figure 8 shows conceptually how a cluster’s neighborhood is partitioned to compute )(,0 αΔD and )(, αλ Δ+Δ gD .

Specifically, gλ is the green noise principal wavelength of the input gray-level g and it is the average distance

between two neighboring clusters. In formulation, it is defined as

gM gg /ˆ=λ (14)

where gM̂ is the average number of minority pixels per cluster for a particular gray level g. In other words, gM̂

is the average cluster size. ∆ is the radius of a circle which is large enough to encircle a cluster for any gray level

g∈(0,0.5]. In our measure, ∆ is selected to be 2.

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The α resolution of )(, αλ Δ+Δ gD doubles that of )(,0 αΔD in terms of number of segments. This is because in a

polar system a region further away from the origin can support a higher pixel resolution.

Intra-cluster distribution:

Figure 9 shows the )(,0 αΔD measures of the outputs of various algorithms for g = 33/255, 60/255, 82/255 and

116/255. Only the upper halves of the plots are shown here as the lower half of the plot of )(,0 αΔD can be

obtained with )(,0 πα +ΔD = )(,0 αΔD . Two observations can be obtained from Figure 9. First, whatever algorithm

is used, the intra-cluster dot distribution is generally more biased to some directions (i.e. The )(,0 αΔD value is

larger for some α’s.) when the input gray-level is low. Second, the distribution of the output of [7] is the most

uneven among the evaluated algorithms and it is remarkably biased to the vertical direction except the case of

g=116/255. On the contrary, though the )(,0 αΔD ’s of the others are also not equal for all α’s, there is no overall

directional bias as their plots are more or less radially symmetric.

To have a complete picture of the directional characteristic of the intra-cluster dot distribution for all input

gray levels, a directional index function based on )(,0 αΔD is defined as

∑=

Δ−=8

1

2,0 ))(1(

81)(

α

αDgDIntra for 0.5≥g>0 (15)

In ideal case, )(gDIntra should be zero for all g because an isotropic distribution of dots makes )(,0 αΔD =1 for all

α. The larger the value, the more severe the distribution is directional for the specific input gray level.

Figure 10 shows the corresponding performance of the evaluated algorithms and from it two observations can

be obtained. First, the intra-cluster dot distribution of the output of [10] is consistent for a wide range of input

gray levels. This is due to the fact that the shape of a cluster generated by [10] is fixed for a wide range of gray

levels as shown in Figure 7(iii). This consistency makes [10] provide the most uniform distribution for most input

gray levels. Second, the performance of FMEDg is comparable to that of [10] when the input gray level g is larger

than 0.2 and can be even better than [10] when g is in the range from 0.38 to 0.48.

Inter-cluster distribution:

Figure 11 shows the )(, αλ Δ+Δ gD plots for the evaluated algorithms. Again, only the upper halves of the plots

are shown here due to the symmetric property of )(, αλ Δ+Δ gD . Two observations can be obtained from Figure 11.

First, though [10] can provide a uniform intra-cluster dot distribution as shown in Figure 10 due to its fixed

predefined cluster characteristic, its inter-cluster dot distribution is severely directional for small g (e.g. g=33/255).

Second, the plot of FMEDg is radially symmetric, which implies there is no directional hysteresis in their

12

halftoning outputs whatever the input gray level is. Moreover, when g is large enough (e.g. g>0.2), its )(, αλ Δ+Δ gD

values are all close to 1 for all α and it is very close to the ideal situation.

To have a complete picture of the directional characteristic of the inter-cluster dot distribution for all input

gray levels, a directional index function based on )(, αλ Δ+Δ gD is defined as

∑=

Δ+Δ−=16

1

2, ))(1(

161)(

αλ α

gDgDInter for 0.5≥g>0 (16)

Similar to )(gDIntra , )(gDInter should be zero for all g in ideal case and a large )(gDInter value indicates a

severely directional characteristic of the inter-cluster dot distribution for the corresponding input gray level g.

Figure 12 shows the corresponding performance of the evaluated algorithms and from it two observations can

be obtained. First, the inter-cluster dot distribution of the output of [10] is highly directional as compared with the

other evaluated algorithms for most input gray levels. This implies the distribution of the clusters in the output of

[10] is also highly directional and there is severe directional hysteresis. Second, the performance of FMEDg is

remarkably better than that of the other evaluated algorithms for all input gray levels. This is expected as a close-

to-isotropic diffusion filter and a not-predetermined scanning path is used in FMEDg.

B. Spectral statistics

Radically averaged power spectrum density (RAPSD) and anisotropy are two measures proposed by Ulichney

to analyze the spectral statistics of a halftone pattern[4]. Both of them are defined based on the power spectrum of

a halftone pattern.

RAPSD

RAPSD is defined as the average power of the frequency components in the annular ring with center radius pf

in the spectral domain as follows.

∑∈

=)(

)(ˆ))((

1)(pfRfp

p fPfRN

fP (17)

where )( pfR is an annular ring of width pΔ partitioned from the spectral domain and ))(( pfRN is the number of

frequency components in )( pfR . )(ˆ fP is the magnitude square of the Fourier transform of the output pattern

divided by the sample size.

Figure 13 shows the performance of various algorithms in terms of RAPSD. A good green noise generator

should produce a result the spectrum of which carries weak low- and high-frequency spectral components and has

13

a spectral peak at green noise principal frequency gf . We can see that all the evaluated algorithms have green

noise characteristics in terms of RAPSD.

Anisotropy

Anisotropy is defined as

∑∈

−

−=

)(2

2

)())()(ˆ(

1))((1)(

pfRf p

p

pp fP

fPfPfRN

fA (18)

It provides the noise-to-signal ratio of frequency samples of )(ˆ fP in )( pfR and is used to measure the strength of

directional artifact.

Figure 14 shows the performance of various algorithms in terms of anisotropy. As mentioned in [4], when

dB0)( >pfA happens, directional components are considered to be strong or noticeable to human eyes. To

provide a reference to study the performance of the algorithms, a surface defined by dB0)( =pfA is added in

each of the plots. The plots show that the proposed FMEDg is better than the other algorithms. Its corresponding

anisotropy is well below 0 dB.

Green noise halftoning is characterized by a distribution of clusters of dots and the distribution should be as

homogeneously as possible [8]. Clusters distributed in this way create an aperiodic and isotropic pattern, which

makes the output visually pleasant as it does not clash with the structure of an image. FMEDg is better in this

aspect.

VI. SIMULATION RESULTS

To study the performance of the evaluated algorithms in handling real images, some 8-bit gray-level testing

images including seven 512×512 natural images shown in Figure 15 and their 256×256 versions were also used in

our simulations. Halftone visibility metrics proposed in [16] were used to measure the distortion observed by a

human viewer between an original gray-level image X and its binary halftone B.

Figure 16 shows the performance of various algorithms in terms of vMSE . In particular, vMSE is defined as

2),,B(),,X(1 dpivdhvsdpivdhvsNN

MSEv −×

= (19)

where hvs is the HVS filter function defined in[16], vd is the viewing distance in inches and dpi is the printer

resolution. In our simulations, the viewing distance changes from 20 to 80 inches and a printer resolution of

600dpi was considered. Table 1 shows the performance in terms of Universal Objective Image Quality Index

(UQI) [17]. Note that the value of UQI is bounded in [-1,1] and a larger value indicates a better performance. One

can see that, in terms of both vMSE and UQI, FMEDg is better than the others.

14

For subjective evaluation, Figure17 shows a portion of the original testing image “Barbara” and Figure18

shows the corresponding outputs of the evaluated algorithms. Several observations can be obtained by comparing

these figures. First, outputs of all evaluated algorithms have green noise characteristics. Second, there is severe

pattern noise in the cheek and the finger area in Figure 18c. Finally, FMEDg preserves more feature details of the

original image than the others. To verify this, one can check the scarf covered the woman’s right shoulder and left

upper arm, and the back of the chair.

The proposed FMEDg allows one to adjust the cluster size continuously and roughly linearly with a single

parameter 1R . Figure 19 shows the halftoning results associated with different 1R values. The noise characteristic

of Figure 19(a) is actually blue while those of the others are green.

VII. COMPUTATIONAL COMPLEXITY

In this section, a computational complexity analysis of the proposed algorithm is provided. The analysis is

based on an assumption that the input image is of size N×N, where N is a value of 2 to the power of a positive

integer.

Obviously, the complexity of FMEDg depends on the number of non-zero coefficients of the diffusion filter,

which is 1R -dependent, and the size of the image. The construction of an energy pyramid at the initialization

stage of FMEDg takes 2N additions. For each iteration cycle, it takes N2log3 comparisons to locate a pixel to

put a dot, multiplications and P additions to diffuse the error, where P is the number of non-

zero diffusion filter coefficients, and ⎡ ⎤∑ −

=

−1log

0

2 4N

ll P 1log3/4 2 −+≈ NP additions to update an energy pyramid

after the diffusion. Here, we assume that no rectification of the diffusion filter is required in the diffusion. Note

that rectification generally results in fewer non-zero filter coefficients.

The value of P can be approximated to be π21R , which is the size of the support area of the ideal ring filter

)2,( 11 RRF . However, this approximation always underestimates the true number. Figure 20 shows how P changes

with 1R in reality. Their relationship can be approximated as 7051.11110.91844.3 12

1 ++= RRP .

For comparison, the complexity for the case using kH , the square filter defined in (1), as the default diffusion

filter is also presented here. In such a case, we have P=8k. Accordingly, for each iteration cycle, it takes N2log3

comparisons to locate a pixel to put a dot, k+1 multiplications and 8k additions to diffuse the error and

⎡ ⎤∑ −

=

−1log

0

2 4N

ll P 1log3/32 2 −+≈ Nk additions to update an energy pyramid. Note that the case of k=1

corresponds to FMED [13].

Table 2 shows the execution time of the Matlab realizations of various green noise error diffusion algorithms

to halftone a 256×256 image for comparison. Diffusion filter oF )28.1,8.1( is used in the realization of FMEDg. The

simulations were carried out on a 2.83GHz Intel Core (TM) 2 PC with 3.25GB RAM. Note that FMEDg does not

15

need to process all N×N pixels. As long as all white dots or black dots are located, B is well-defined and FMEDg

can be terminated. Accordingly, the execution time of FMEDg is image dependent.

The execution time of FMEDg reported in the table reflects the complexity of its straight-forward Matlab

implementation without any optimization. An optimized C realization can effectively reduce the execution time.

As a matter of fact, the complexity of MED algorithms is generally much higher than that of conventional

error diffusion algorithms. However, arithmetic operations can be reduced by recycling the intermediate

processing results. Besides, the processing time can be shortened by localizing the data processing operations to

reduce the computation effort and support parallel processing. The successful example of complexity reduction

reported in [14] shows that one can achieve blue noise halftoning with MED at a complexity of 17 arithmetic

operations per pixel for a 256×256 image. Similar idea can be used to reduce the computational complexity of

FMEDg.

In this paper, the focus is put on the halftone quality and the cluster adjustability. The fast realization of

FMEDg is to be explored in the future.

VIII. CONCLUSIONS

MED algorithms are able to produce halftones of good blue noise characteristics without directional hysteresis.

When the dot gain of a printing process is not stable, this capability may not be useful as a halftone of green noise

characteristics is expected in such a case. In this paper, we proposed a method to make FMED [13] produce

halftones of good green noise characteristics. Specifically, a diffusion filter is designed based on an isotropic ring

filter with their inner and outer radii ( 1R and 2R ) as adjustable parameters. FMED can then be modified to work

with the proposed filter to achieve green noise halftoning.

A detailed empirical analysis was made and we show that, when 2/ 12 =RR , the connection of the average

dot cluster size gM , the inner radius 1R and the input gray level g can be modeled with gRM g π21= .

Consequently, the proposed algorithm allows one to select any desirable average dot cluster size for a particular

input gray level based on the dot gain nature of a printing process easily and freely by adjusting parameter 1R .

This flexible adjustability and the linear relationship between gM and g are two advantageous features of the

proposed algorithm to provide halftones of desirable green noise characteristics.

Empirical performance analysis results show that the proposed algorithm can provide a better performance as

compared with the other green noise error diffusion algorithms in terms of dot distribution, anisotropy, and

various image quality measures including vMSE and UQI. The proposed algorithm eliminates pattern artifacts

and directional hysteresis completely, and preserves image details better as compared with the others.

16

APPENDIX

In general, for any pixel belongs to {(m,n)|m ≥0,n>0}, the coverage of kRyx ≤+ 22 must be in one of the

eight scenarios shown in Figure A1. The classification of cases 3 to 8 is based on the intersections of the circle

and the grid. The criteria are given as follows:

case 1: kRnm ≥−+− 22 )5.0()5.0( if m>0 and kRn ≥− 5.0 if m=0

case 2: kRnm ≤+++ 22 )5.0()5.0(

case 3: 5.05.0 +<<− nnn l and 5.05.0 +<<− mmm l

case 4: 5.05.0 +<≤− nnn h and 5.05.0 +<≤− mmm h

case 5: 5.05.0 +<<− nnn l and 5.0 5.0 +<≤− nnn h

case 6: 5.05.0 +<<− mmm l and 5.0 5.0 +<≤− mmm h

case 7: 5.0≤lm and m=0

case 8: 5.0≤hm and m=0

where 22 )5.0( +−= mRn kh , 22 )5.0( −−= mRn kl , 22 )5.0( +−= nRm kh and 22 )5.0( −−= nRm kl .

Accordingly, ),,( kRnmA can be determined as

( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

+

+−−+−

++−−−

=

8),5.0,(27),,0(26),,()5.0(5),5.0,5.0(4),5.0,()5.0(3),,5.0(2110

),,(

caseifnmfmcaseifnmfcaseifnmmfmmcaseifnmmfcaseifnmmfmmcaseifnmmfcaseifcaseif

RnmA

hAh

lA

lhAh

A

hAh

lA

k0,0for >≥ nm (A1)

where

( )dxnxRnqpfq

pkA ∫ −−−= )5.0(),,( 22 (A2)

),,( kRnmA for (m,n)’s in other quadrants can be determined by making use of the following symmetric

property

),,(),,( kk RnmARnmA = (A3)

17

REFERENCES

[1] R. A. Ulichney, Digital Halftoning. Cambridge, MA:MIT Press, 1987.

[2] J. C. Stoffel and J. F. Moreland, “A survey of electronic techniques for pictorial reproduction,” IEEE Trans. Communication. 29, 1898–1925, 1981.

[3] R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. S.I.D. 17(2), 75–77, 1976.

[4] R. A. Ulichney, “Dithering with blue noise,” Proc. IEEE, vol. 76, pp. 56–79, Jan. 1988.

[5] B. Kolpatzik and C. A. Bouman, “Optimized error diffusion for image display,” Journal of Electronic Imaging, 1(3), 277–292, 1992.

[6] T. N. Pappas and D. L. Neuhoff, “Printer models and error diffusion,” IEEE Trans. Image Processing, vol. 4, pp. 66–79, Jan. 1995.

[7] R. Levien, “Output dependent feedback in error diffusion halftoning," IS&T Imaging Science and Technology 1, pp. 115-118, May 1993.

[8] D. L. Lau, G. R. Arce, and N. C. Gallagher, “Green-noise digital halftoning,” Proceedings of the IEEE 86, pp. 2424-2442, Dec. 1998

[9] N. Damera-Venkata and B. L. Evans, “Adaptive threshold modulation for error diffusion halftoning,” IEEE Trans. Image Process., vol. 10, no. 1, pp. 104–116, Jan. 2001.

[10] N. Damera-Venkata, J. Yen, V. Monga and B. L. Evans, “Hardcopy Image Barcodes Via Block Error Diffusion,” IEEE Trans. on Image Processing, vol. 14, no.12, pp.1977-1989, 2005.

[11] I. Katsavounidis and C. C. J. Kuo, “A multiscale error diffusion technique for digital halftoning,” IEEE Trans. on Image Processing. Vol.6, No.3, pp.483–490,1997.

[12] Y.H. Chan, “A modified multiscale error diffusion technique for digital halftoning,” IEEE Signal Processing Letters, vol. 5, no.11, pp. 277-280, 1998.

[13] Y.H. Chan and S. M. Cheung, “Feature-preserving multiscale error diffusion for digital halftoning,” Journal of Electronic Imaging, 13(3), pp. 639-645, 2004.

[14] Y.H. Fung, K.C. Lui and Y.H. Chan, “low-complexity high-performance multiscale error diffusion technique for digital halftoning,” Journal of Electronic Imaging, 16(1), (2007).

[15] Y.H. Fung and Y.H. Chan, “Embedding halftones of different resolutions in a full-scale halftone,” IEEE Signal Processing Letters, vol. 13, no.3, pp. 153-156, 2006.

[16] D. L. Lau and G. R. Arce, Modern Digital Halftoning, Marcel Dekker, New York NY, USA 2001.

[17] Zhou Wang and Alan C. Bovik, “A Universal Image Quality Index,” IEEE Signal Processing Letters, vol. 9, no.3, pp.81-84, 2001.

18

Figure Caption

Figure 1. How FMED[13] operates in an iteration cycle

Figure 2. The conceptual model of the cluster formation with an ideal ring filter in continuous spatial domain.

Figure 3. The frequency responses of (a) square filter 2H , (b) approximated ring filter oF)28.1,8.1( and (c) ideal

ring filter )28.1,8.1(

F .

Figure 4. Variance of the frequency components covered in an annular ring of any radius r. (a) R1=1.8 and (b) R1=2.8.

Figure 5. Relationship between inner radius R1, input gray level and cluster size when 2/ 12 =RR . (a) Average of the cluster sizes obtained in the simulation, gM̂ ; (b) standard deviation of the cluster sizes

obtained in the simulation; (c) cluster size derived by the model, gM ; and (d) difference between gM̂ and gM .

Figure 6. Average measured cluster size obtained with different combinations of input gray level and inner radius 1R when (a) 12 1213.2 RR = , (b) 12 8284.1 RR = , (c) 12 2RR = and (d) 12 1.1 RR = .

Figure 7. Parts of the halftoning results of a 256×256 input of constant gray-level (a) g=33/255, (b) g=60/255, (c) g=82/255 and (d) g=116/255 with (i) Levien [7], (ii) Damera-Venkata [9], (iii) Damera-Venkata [10] and (iv) FMEDg.

Figure 8. How intra- and inter-cluster dot distributions are measured with )(,0 αΔD and )(, αλ Δ+Δ gD for a

particular input gray level g.

Figure 9. Directional distribution )(,0 αΔD plots for the halftoning results of a 256×256 input of constant gray-level (a) g=33/255, (b) g=60/255, (c) g=82/255 and (d) g=116/255 with (i) Levien [7], (ii) Damera-Venkata [9], (iii) Damera-Venkata [10] and (iv) FMEDg.

Figure 10. Intra-cluster distribution performance

Figure 11. Directional distribution )(, αλ Δ+Δ gD plots for the halftoning results of a 256×256 input of constant

gray-level (a) g=33/255, (b) g=60/255, (c) g=82/255 and (d) g=116/255 with (i) Levien [7], (ii) Damera-Venkata [9], (iii) Damera-Venkata [10] and (iv) FMEDg.

Figure 12. Inter-cluster distribution performance

Figure 13. Performance in terms of RAPSD: (a) Levien [7], (b) Damera-Venkata [9], (c) Damera-Venkata [10] and (d) FMEDg.

Figure 14. Performance in terms of anisotropy: (a) Levien [7], (b) Damera-Venkata [9], (c) Damera-Venkata [10] and (d) FMEDg.

Figure 15. Testing Images

Figure 16. The average MSEv of the testing images of size (a) 256×256 and (b) 512×512 at different viewing distances (printer resolution = 600dpi).

Figure 17. A cropped region of original “Barbara”

Figure 18. Halftones produced with various algorithms

Figure 19. FMEDg outputs obtained with (a) 1R = 0.8, (b) 1R = 1.3, (c) 1R = 1.8 and (d) 1R = 3.0.

Figure 20. Number of non-zero coefficients of ring filter oRRF )2,( 11

.

Figure A1. Area covered by circle kRyx ≤+ 22 in a grid unit in the first quadrant.

19

Table caption

Table 1. UQI performance of various algorithms

Table 2. The execution time of various algorithms

20

Figure 1. How FMED[13] operates in an iteration cycle

Figure 2. The conceptual model of the cluster formation with an ideal ring filter in continuous spatial domain.

21

(a) (b) (c)

Figure 3. The frequency responses of (a) square filter 2H , (b) approximated ring filter oF)28.1,8.1( and (c) ideal ring filter

)28.1,8.1(F .

(a) (b) Figure 4. Variance of the frequency components covered in an annular ring of any radius r. (a) R1=1.8 and (b) R1=2.8.

22

(a) (b)

(c) (d)

Figure 5. Relationship between inner radius R1, input gray level and cluster size when 2/ 12 =RR . (a) Average of the

cluster sizes obtained in the simulation, gM̂ ; (b) standard deviation of the cluster sizes obtained in the

simulation; (c) cluster size derived by the model, gM ; and (d) difference between gM̂ and gM .

(a) (b)

(c) (d)

Figure 6. Average measured cluster size obtained with different combinations of input gray level and inner radius 1R

when (a) 12 1213.2 RR = , (b) 12 8284.1 RR = , (c) 12 2RR = and (d) 12 1.1 RR = .

23

(i)

(ii)

(iii)

(iv)

(a) (b) (c) (d) Figure 7. Parts of the halftoning results of a 256×256 input of constant gray-level (a) g=33/255, (b) g=60/255, (c) g=82/255

and (d) g=116/255 with (i) Levien [7], (ii) Damera-Venkata [9], (iii) Damera-Venkata [10] and (iv) FMEDg.

24

Figure 8. How intra- and inter-cluster dot distributions are measured with )(,0 αΔD and )(, αλ Δ+Δ gD for a particular input

gray level g.

.

25

(i)

(ii)

(iii)

(iv)

(a) (b) (c) (d) Figure 9. Directional distribution )(,0 αΔD plots for the halftoning results of a 256×256 input of constant gray-level (a)

g=33/255, (b) g=60/255, (c) g=82/255 and (d) g=116/255 with (i) Levien [7], (ii) Damera-Venkata [9], (iii) Damera-Venkata [10] and (iv) FMEDg.

Figure 10. Intra-cluster distribution performance

26

(i)

(ii)

(iii)

(iv)

(a) (b) (c) (d) Figure 11. Directional distribution )(, αλ Δ+Δ g

D plots for the halftoning results of a 256×256 input of constant gray-level (a) g=33/255, (b) g=60/255, (c) g=82/255 and (d) g=116/255 with (i) Levien [7], (ii) Damera-Venkata [9], (iii) Damera-Venkata [10] and (iv) FMEDg.

Figure 12. Inter-cluster distribution performance

27

(a) (b)

(c) (d)

Figure 13. Performance in terms of RAPSD: (a) Levien [7], (b) Damera-Venkata [9], (c) Damera-Venkata [10] and (d) FMEDg.

(a) (b)

(c) (d) Figure 14. Performance in terms of anisotropy: (a) Levien [7], (b) Damera-Venkata [9], (c) Damera-Venkata [10] and (d)

FMEDg.

28

House Lena Mandrill Barbara

Boat Peppers Man

Figure 15. Testing Images

(a) (b)

Figure 16. The average MSEv of the testing images of size (a) 256×256 and (b) 512×512 at different viewing distances (printer resolution = 600dpi).

29

Figure 17. A cropped region of original “Barbara”

(a) Levien [7] (b) Damera-Venkata [9]

(c) Damera-Venkata [10] (d) Proposed FMEDg

Figure 18. Halftones produced with various algorithms

30

(a) =1R 0.8 (b) =1R 1.3 (c) =1R 1.8 (d) =1R 3.0

Figure 19. FMEDg outputs obtained with (a) 1R = 0.8, (b) 1R = 1.3, (c) 1R = 1.8 and (d) 1R = 3.0.

Figure 20. Number of non-zero coefficients of ring filter o

RRF )2,( 11.

31

Figure A1. Area covered by circle kRyx ≤+ 22 in a grid unit in the first quadrant.

32

UQI Image

size Testing Image [7] [9] [10] FMEDg

Mandrill 0.1402 0.1402 0.1233 0.2089 Barbara 0.0972 0.0957 0.0946 0.1436

Boat 0.0713 0.0685 0.0624 0.0979 House 0.0498 0.0487 0.0414 0.0879 Lena 0.0500 0.0496 0.0438 0.0721 Man 0.0638 0.0623 0.0563 0.0944

Peppers 0.0512 0.0511 0.0478 0.0656 51

2×51

2 Average 0.0748 0.0737 0.0671 0.1101 Mandrill 0.1073 0.1077 0.0927 0.1777 Barbara 0.0981 0.0976 0.0960 0.1447

Boat 0.1074 0.1037 0.0962 0.1388 House 0.0698 0.0681 0.0573 0.1140 Lena 0.0857 0.0846 0.0756 0.1087 Man 0.0918 0.0902 0.0809 0.1297

Peppers 0.1040 0.1028 0.0955 0.1194

256×

256

Average 0.0949 0.0935 0.0849 0.1333

Table 1. UQI performance of various algorithms

Time (second) Testing

Image [7] [9] [10] FMEDg Mandrill 0.67 1.25 1.00 181.84 Barbara 0.66 1.23 0.98 165.13

Boat 0.66 1.23 0.98 226.75 House 0.66 1.25 0.98 117.63 Lena 0.66 1.24 0.97 219.11 Man 0.66 1.25 0.99 156.39

Peppers 0.64 1.23 0.99 154.64

Table 2. The execution time of various algorithms.